The method of lie series in the motion-separation problem in nonlinear mechanics

Abstract

A method is proposed for separating the motions for multifrequency systems in standard form, based on the construction of the Hamiltonian form of the system by introducing adjoint variables, and subsequent use of the group of canonical transformations generated by a Lie generator. The method yields a simpler algorithm for constructing higher approximations compared with the well-known Krylov-Bogolyubov method /1,2/, since the transformation of n + m equations is replaced by a transformation of one scalar function, while instead of the n + m equations of the change of variables a scalar equation is set up for the Lie generator and its partuclar solution is derived in closed form. It is shown that the introduction of adjoint variables does not lead to an increase in the dimensions of the problem. A definition of resonance in the i-th approximation is introduced and a resonance form of the method is given. An example is presented.